Abstract

An overview of the theory of the magneto-optical trap is presented, along with measurements of the effect of an imbalance in the intensities of the trapping beams. This investigation tests the theory of the spring constant of the trap and confirms that the confining force at the center of the trap results from an induced orientation of the atomic ground state. The experimental results give the magnitude of this force, which has not yet been calculated accurately. We calculate the radiation field in the three-dimensional molasses, finding that the relative time phase of the orthogonal standing waves is significant, and we give some insight into the phenomenon of interference fringes when the beams are misaligned. We also discuss the limitation of the trapped atomic density resulting from photon scattering within the cloud, predicting that densities above 1013 atoms/cm3 could be achieved in a trap operating at low saturation of the atomic transition. Finally, we briefly consider collisional loss at low densities, finding an especially large contribution from resonant dipole–dipole scattering.

We use this expression following Refs. 28 and 29, although the name is slightly misleading, since there is no oscillating magnetic field: The mathematical description is merely analagous to a magnetic resonance.

We use this expression following Refs. 28 and 29, although the name is slightly misleading, since there is no oscillating magnetic field: The mathematical description is merely analagous to a magnetic resonance.

Figures (10)

Presence of Sisyphus cooling in the trap. (a)–(c) show the parameter p defined in the text. The calculation added six plane waves of unit amplitude, each pair having opposite circular polarizations, as for a trap. (a) Aligned beams, all at the same phase angle. (b) Aligned beams, with the x beam pair shifted in phase by π/2 rad with respect to the others. (c) As in (a) but with the x beams misaligned in the x–y plane, one by +0.05 rad, the other by −0.05 rad. Each graph shows the variation of p along a set of four lines parallel to the y axis (which is also the quantization axis), the lines being separated in the x direction by a quarter of a wavelength. For some cases p = 0 is obtained for all y. (d) Intensity for the misaligned-beams case. The wavelength-scale intensity fluctuations occur along the same lines for which circular polarization is present but at different y positions. All the graphs are at z = 0. At z = +0.25 wavelengths, we still find variations in p similar to those shown in (c), but the intensity in (d) fluctuates between approximately 2 and 10 for all x and y. The wavelength-averaged optical potential has an almost uniform depth throughout the trap.

Various significant radii in the trap. Top, the radius of the cloud, as given by relation (57) (3D) and by a calculation in which all the parameters are taken from the one-dimensional induced-orientation theory (1D). Bottom, radii at which the Larmor frequency is equal to the optical pumping rate and to the light shift. Note the different vertical scales. The example given is for Ω = Γ/2 and a field gradient of 10 G/cm.

Inverse of the trap lifetime as a function of background pressure. The trapping light was at 6-MHz detuning, 7 mW/cm2 per beam. The experimental uncertainty in the lifetime measurements is indicated by the size of the symbols. The pressure given is that indicated by the ion gauge (calibrated for N2). If the background gas is cesium, the actual pressure is less than that indicated by the gauge.

Imbalance measurements. All the data are shown. The vertical axis shows the magnetic field, in gauss, to which the trapped cloud moves, per unit intensity imbalance w in the vertical beams. The straight line is a fit to the data points for which Ω2/δΓ < 0.4. The symbols indicate the intensity per beam as follows: I/IS = 2.3, ●; 2, ◊; 1.7, ○; 1.4, □; 1, ▲; 0.7, ∇; 0.55, Δ.

Imbalance measurements as in Fig. 7, rescaled by the expected detuning dependence in order to reveal the dependence on the average intensity per trapping beam. Only the data points for detunings larger than 10 MHz and Ω2/δΓ smaller than 0.4 are included. The line is a linear fit to the data.

Equations (74)

(3)Number of atoms in the trapN,Number density of atomsn,Lifetime for an atom to remain in the trapτ,Maximum velocity for an atom to be captured by the trapυc,Escape velocity for an atom in the center of the trapυesc.